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### Proyecciones (Antofagasta) vol.37 no.1 Antofagasta Mar. 2018

#### http://dx.doi.org/10.4067/S0716-09172018000100019

Articles

Vertex graceful labeling of some classes of graphs

1 Hindustan Institute of Technology and Science, Department of Mathematics, Tamil Nadu, India. email : apskumar1953@gmail.com

2 Hindustan Institute of Technology and Science, Department of Mathematics, Research Scholar, Tamil Nadu, India. email : balaganesanpp@gmail.com

Abstract:

A connected graph G = (V,E) of order atleast two, with order p and size q is called vertex-graceful if there exists a bijection ʄ : V → { 1, 2, 3, ··· p } such that the induced function ʄ*: E → { 0, 1, 2, ··· q-1} defined by ʄ*(uv) = (ʄ(u)+ ʄ(v)) (mod q) is a bijection. The bijection ʄ is called a vertex-graceful labeling of G. A subset S of the set of natural numbers N is called consecutive if S consists of consecutive integers. For any set X, a mapping ʄ : X → N \$ is said to be consecutive if ʄ(X) is consecutive. A vertex-graceful labeling ʄ is said to be strong if the function ʄ1: E → N defined by ʄ1(e)= ʄ(u) + ʄ(v) for all edges e = uv in E forms a consecutive set. It is proved that one vertex union of odd number of copies of isomorphic caterpillars is vertex-graceful and any caterpillar is strong vertex-graceful. It is proved that a spider with even number of legs (paths) of equal length appended to each vertex of an odd cycle is vertex-graceful. It is also proved that the graph lA(mj,n) is vertex-graceful for both n and l odd, 0 ≤ i ≤ n-1, 1 ≤ j ≤ mi. Further, it is proved that the graph A(mj, n) is strong vertex-graceful for n odd, 0 ≤ i ≤ n-1, 1 ≤ j ≤ mi.

Key words: Caterpillar; one vertex union graphs; regular spider; actinia graph; vertex-graceful labeling; strong vertex-graceful labeling

1. Introduction

By a graph G= (V,E), we mean a finite undirected graph without loops or multiple edges. The order and size of G are denoted by p and q, respectively. For basic graph theoretic terminology, we refer to (5) (8). A caterpillar is a tree, the removal of whose end vertices, gives a path. The resulting path is called a spine. Denote by C(n,mi), a caterpillar, where n is the number of vertices on the spine and mi is the number of end vertices adjacent to each vertex of the spine. Let Gi(1 ≤ i ≤ l)\$ be a connected graph with vertex set Vi= {v1 i, v2 i, v3 i, ···, vnl i} . Then the one vertex union of Gi(1≤ i ≤ l) at v1 1 is obtained by identifying all the vertices v1 1, v1 2, v1 3, ···, v1 l at v1 1 and is denoted by G(l).

A spider is a tree that has at most one vertex (called the center) of degree greater than 2. A regular spider is the one vertex union of paths of equal length. A bamboo tree is a rooted tree consisting of branches of equal length, the end points of which are identified with the end points of stars of different sizes. A bamboo tree is called regular if the sizes of the stars considered are equal. The bamboo trees were introduced and studied in (11). The graph A(mi,n) obtained by identifying mi, (1≤ i ≤ n) pendant edges to the vertices of a cycle Cn is called Actinia graph. The Actinia graphs were introduced and studied in (19).

A graph labeling is an assignment of numbers to the vertices or edges, or both vertices and edges subject to certain conditions. A graph G is called graceful graph if there exists an injection ʄ : V → {0,1,2, ···, q} such that the induced function ʄ* : E → { 1,2, ···, q} defined by ʄ*(uv)= | ʄ(u)- ʄ(v)| is a bijection. The injection ʄ is called a graceful labeling of graph G. The values ʄ(u) and ʄ*(uv) are called the graceful labels of the vertex u and the edge uv, respectively. A graph G is called vertex-graceful if there exists a bijection ʄ : V → { 0,1,2, ···, p } such that the induced function ʄ* : E → { 0, 1,2, ···, q-1 } by ʄ*(uv) = (ʄ(u) + ʄ (v))(mod q) is a bijection. The bijection ʄ is called a vertex-graceful labeling of G. Let N denote the set of natural numbers. A subset S of N is called consecutive if S consists of consecutive integers. For any set X, a mapping ʄ : X → N is said to be consecutive if ʄ(X ) is consecutive. A vertex-graceful labeling ʄ is said to be strong if the function by ʄ1: E → N defined by ʄ1(e) = ʄ(u) + ʄ(v) for all edges e=uv in E forms a consecutive set.

In 1967, Rosa (12) introduced the labeling method called β-valuation. A function ʄ is called β-valuation of a graph G with q edges if ʄ is an injection from the vertices of G to the set { 0,1,2 ,···, q} such that, when each edge xy is assigned the label | ʄ(x)- ʄ(y)|, the resulting edge labels are distinct. Golomb (7) called such labelings graceful and this term is followed presently. The book edited by Acharia, Arumugam and Rosa (1) includes a variety of labeling methods. Hsu and Keedwell (9) (10) introduced and studied the extension of graceful labeling of directed graphs. The relationship between graceful directed graphs and a variety of algebraic structures including cyclic difference sets, sequenceable groups, generalized complete mappings, near-complete mapping and neofield is discussed in (3) and (4). Bahl, Lake and Wertheim (2) proved that spiders for which the lengths of every path from the center to a leaf differ by at most one are graceful and spiders for which the lengths of every path from the center to a leaf has the same length and there is an odd number of such paths there is a family of graceful labelings.

Graph labeling, where the vertices are assigned values subject to certain conditions have often been motivated by practical problems. Labeled graphs serve as useful mathematical models for a broad range of applications such as coding theory, including the design of good radar type codes, synch-set codes, missile guidance codes and convolution codes with optimal autocorrelation properties. The notion of vertex-graceful graph of order p and size p+1 was introduced and studied in(18). The vertex graceful labeling of many classes of graphs were studied in (13) (14) (15) (16) (17). For detailed study of labeling of graphs, we refer to (6).

2. Vertex Gracefulness of One Vertex Union of Caterpillars

In this section, it is proved that one vertex union of odd number of copies of isomorphic caterpillars is vertex-graceful and also proved that any caterpillar is strong vertex graceful.

Let G=C(l)(n,mi) be the one vertex union of l copies of isomorphic caterpillars with n vertices in the spine and each of its vertices are appended with mi (1 ≤ i ≤ n) pendant vertices. Let v1 k, v2 k,v3 k, ···, vn k be the vertices of the spine of the kth copy of the isomorphic caterpillars. Let m1, m2, m3, ···, mn be the number of pendant vertices, which are appended to each vertex of the spine of the caterpillar. Let v1 1, v1 2,v1 3, ···, v1 l be the initial vertex of each spine of l isomorphic caterpillars. The one vertex union of l copies of isomorphic caterpillars at v1 1 is obtained by identifying all the vertices v1 1, v1 2,v1 3, ···, v1 l at v1 1. Let vij k be the pendant vertices of G for 1 ≤ i ≤ n, 1 ≤ j ≤ mi, 1 ≤ k ≤ l. This graph is shown in Figure 2.1.

Theorem 2.1. For 1 ≤ i ≤ n, 1 ≤ j ≤ mi and l odd, the graph G=C(l)(n,mi) is vertex-graceful.

Proof. It is clear from the definition of G that the number of vertices p and edges q are

p=l +1 and q=p-1. Let α =n-1+ . The required vertex-graceful labeling for vertices is defined as follows:

Let ʄ: V → {1,2,···, p}.

Case 1. n even.

Let β =

Define ʄ(v1 1)= 1,

The vertex labeling of pendant vertices at v1 k, v2 k,v3 k, ···, vn k is given by

The corresponding edge labeling is defined as follows:

It is clear that all the vertex labels are distinct. Also it is easily verified that the edge label sets A, B, C, D and E are mutually disjoint and ABCDE = {0,1,2, ···, q-1}. Therefore, it follows that the induced mapping ʄ* : E → {0,1,2, ···, q-1} is a bijection. Hence the graph G is a vertex-graceful when n is even.

Case 2. n odd.

The vertex labeling of pendant vertices at v1 k, v2 k,v3 k, ···, vn k is given by

The corresponding edge labeling is defined as follows:

It is clear that all the vertex labels are distinct. Also it is easily verified that the edge label sets A, B, C, D and E are mutually disjoint and ABCDE = {0,1,2, ···, q-1}. Therefore, it follows that the induced mapping ʄ*: E → {0,1,2, ···, q-1} is a bijection. Hence the graph G is a vertex-graceful when n is odd. Thus the graph G is vertex-graceful.

Illustrative examples of the labeling of the graph G in Theorm 2.1 are given in Figures 2.2 and 2.3.

We leave the following problem as an open question.

Problem 2.2. Is it true that for 1 ≤ i ≤ n, 1 ≤ j ≤ mi and l even, the graph G=C(l)(n,mi) is vertex-graceful?

Remark 2.3. If l=1 in Figure 1, then the graph G represents a caterpillar. Hence it follows from Theorem 2.1 that any caterpillar is vertex-graceful. Also if mi=0 for i=1, 2,···,n-1 and mn ≠0 in Figure 2.1, then the graph G represents a regular bamboo tree. Hence it follows from Theorem 2.1 that a regular bamboo tree is vertex-graceful.

Theorem 2.4. Any caterpillar G is strong vertex graceful.

Proof. Let the caterpillar G be a tree with n vertices v1, v2, ···, vn on the spine and each vi is attached with mi pendant edges (1 ≤ i ≤ n). The graph is given in Figure 2.4.

Then p=n+ and q=n-1+ . The required strong vertex-graceful labeling for vertices is defined as follows:

Let ʄ: V → {1,2, ··· ,p} be such that

Case 1. n even.

Let

The vertex labeling of pendant vertices at v1, v2, v3, ···, vn is given by

The corresponding edge labeling is defined as follows:

It is clear that the vertex labels are distinct. Also it is easily verified that the edge labels sets are mutually disjoint and A ∪ B = . Therefore, the induced edge labels of G have q consecutive values. Hence the graph G is strong vertex-graceful when n is even.

Case 2. n odd.

Let γ =

The vertex labeling of pendant vertices at v1, v2, v3, ···, vn is given by

The corresponding edge labeling is defined as follows:

It is clear that the vertex labels are distinct. Also it is easily verified that the edge labels sets are mutually disjoint and A ∪ B = . Therefore the induced edge labels of G have q consecutive values. Hence the graph G is strong vertex-graceful when n is odd. Thus the graph G is strong- vertex graceful.

3. Vertex-graceful Labeling of Cycle with Regular Spider

In this section, it is proved that a spider with even number of legs (paths) of equal length appended to each vertex of an odd cycle is vertex-graceful.

Definition 3.1. A subdivision of an edge uv in a graph is obtained by removing edge uv, adding a new vertex w and adding edges uw and vw. A (wounded) spider is the graph formed by subdividing at most t-1 of the edges of a star K1,t for t ≥ 0.

Examples of wounded spiders include K1, the star K1,n-1, and the graph shown in Figure 3.1.

Definition 3.2. A regular spider is the one vertex union of paths of equal length. Denote by CnPn (l) a cycle Cm (m ≥ 3) attached at each of its vertices a regular spider Pn (l), where l denotes the number of paths Pn in the spider (Refer Figure 3.2).

Example 3.3.

Theorem 3.4. For any n, the graph CnPn (l) is vertex-graceful where m odd and l even.

Proof. Let v0, v1, ···, vm-1 be the vertices of the cycle Cm. Now, let CnPn (l) be the graph as shown in Figure 6. It is easily verified that G has p=m vertices and q=p edges. Let α = l(n-1)+1. The required vertex-graceful labeling for vertices is defined as follows :

Let ʄ: V → {1,2,···, p} be such that

Case 1. n even.

The corresponding edge labeling is defined as follows:

It is clear that the vertex labels are distinct. Also it is easily verified that the edge label sets A, B, C, D and E are mutually disjoint and ABCDE = {0,1,2, ···, q-1}. Therefore, it follows that the induced mapping ʄ*: E → {0,1,2, ···, q-1} is a bijection. Hence the graph G is vertex-graceful when n is even.

Case 2. n odd.

The corresponding edge labeling is defined as follows:

It is clear that the vertex labels are distinct. Also it is easily verified that the edge label sets A, B, C, D and E are mutually disjoint and ABCDE = {0,1,2, ···, q-1}. Therefore, it follows that the induced mapping ʄ* : E → {0,1,2, ···, q-1} is a bijection. Hence the graph G is a vertex-graceful when n is odd.

Illustrative example of labeling of Theorem 3.1 is given in Figures 3.3 and 3.4.

We leave the following problem as an open question.

Problem 3.5. For any n, is it true that the graph CnPn (l) is vertex-graceful when

(i) m odd and l odd ?

(ii) m even and l odd ?

(iii) m even and l even ?

4. Vertex-graceful Labeling of l copies of Actinia Graphs

In this section, it is proved that l copies of the actinia graphs A(mj,n) is vertex-graceful. Let v0 k, v1 k, v2 k, ···, vk n-1 be the vertices of kth copy of the cycle in lA(mj,n) for 0 ≤ j ≤ n-1. Let vij k denote the kth pendant vertices of the kth copy of lA(mj,n) such that vij k are adjacent to vertices vi k for 0 ≤ i ≤ n-1, 1≤ j≤ mi, 1 ≤ k ≤ l. The graph lA(mj,n) is shown in Figure 4.1.

Theorem 4.1. The graph lA(mj,n) is vertex-graceful for both n and l odd, 0 ≤ i ≤ n-1, 1 ≤ j ≤ mi.

Proof. Consider the graph G=lA(mj,n) for both n,l odd, 0 ≤ i ≤ n-1, 1 ≤ j ≤ mi. Let v0 k, v1 k, v2 k, ···, vk n-1 be the vertices of the kth copy of the cycle in G. Let vij k denote the pendant vertices of the kth copy such that vij k are adjacent to the vertices vi k for 0 ≤ i ≤ n-1, 1≤ j≤ mi, 1 ≤ k ≤ l. It is easily verified that G has p=l vertices and q=p edges. The required vertex-graceful labeling for vertices is defined as follows:

Let ʄ: V → {1,2,···,p} be such that

The corresponding edge labeling is defined as follows:

It is clear that the vertex labels are distinct. Also it is easily verified that the edge labels sets are mutually disjoint and AB = {0,1,2, ···, q-1} . Therefore it follows that the induced mapping ʄ* : E → {0,1,2, ···, q-1} is a bijection. Hence the graph lA(mj,n), 0 ≤ i ≤ n-1, 1 ≤ j ≤ m_i is vertex-graceful for both n and l odd.

We leave the following problem as an open question.

Problem 4.2. Is it true that the graph lA(mj,n) is vertex-graceful when

(i) n odd and l even ?

(ii) n even and l even ?

(iii) n even and l odd ?

Illustrative example of the labeling of G of Theorem 4.1 is given in Figure 4.2.

Remark 4.3. If l=1 in Figure 4.1, then the graph G represents a actinia. Hence it follows from Theorem 4.1 that the graph A(mj,n) is vertex-graceful for any mj, 0 ≤ j ≤ n-1, n ≥ 3 with n odd.

Theorem 4.4. The graph A(mj,n) is strong vertex-graceful for n odd, 0 ≤ i ≤ n-1, 1 ≤ j ≤ mi .

Proof. Consider the graph G=A(mj,n) for n odd, 0 ≤ i ≤ n-1, 1 ≤ j ≤ mi. Let v0, v1, v2, ···, vn-1 be the vertices of the cycle in G. Let vij denote the pendant vertices of G such that vij are adjacent to the vertices vi for 0 ≤ i ≤ n-1, 1≤ j≤ mi. It is easily verified that G has p=n+ vertices and q=p edges. The required strong vertex-graceful labeling for vertices is defined as follows:

Let ʄ: V → {1,2,···,p} be such that

The corresponding edge labeling is defined as follows:

It is clear that the vertex labels are distinct. Also it is easily verified that the edge labels sets are mutually disjoint and AB = {β+2, β+3, ···, β+m0+m1+···,+mn-2+mn-1+n+1}. Therefore the induced edge labels of G have q consecutive values. Hence the graph G is strong vertex-graceful.

Illustrative examples of the labeling of Theorem 4.4 is given in Figure 4.3.

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Received: July 2016; Accepted: November 2016